family balancing
pp. Attempting to select the sex of a baby to achieve an even or desired number of children of both sexes.
The Laguna Hills clinic, which is a joint effort with the Huntington Reproductive Center of Southern California, offers sex selection for two purposes: to help couples avoid passing on a gender-linked genetic disease and to allow families who already have a child of one gender "balance" their families by having a baby of the opposite sex.

The technology is still experimental; couples using the method will do so as part of a clinical trial overseen by the Food and Drug Administration. However, Blauer says the company has impressive success rates; 91% of the women who become pregnant after sorting for a girl are successful, while 76% who sort for a boy and get pregnant are successful. …

While the Meissners wished to avoid passing on a genetic disease, 86% of the MicroSort's clients so far have used the method for "family balancing," with most of them seeking girls.
—Shari Roan, “A way to choose a baby's gender,” Los Angeles Times, March 03, 2003
Comhaire, a quietly spoken Belgian with excellent English, offers 'family balancing', in which a couple with a girl or a boy can ensure that their next child is of the opposite sex.
—Jo Revill, “Investigation: Will it be a girl or a boy?,” The Observer, September 08, 2002
1994 (earliest)
But for every new medical achievement there are potentially troubling issues. It is now possible through preimplantation genetic testing to screen for devastating hereditary diseases such as Tay-Sachs and cystic fibrosis. But what about someday screening for conditions like blindness or deafness, which are hardly life threatening but nevertheless affect the quality of life? Who determines whether embryos with genetic defects should be used or not? Is it wrong to allow only perfect babies to be born?

How about screening for the "right" sex? Dr. Schulman of the Genetics & IVF Institute says his clinic does not offer gender selection. But he says he can see its application in "family balancing." Says Schulman: "If a family has three children of one sex, would it really be so wrong to help bring about the birth of the opposite sex?" No doubt plenty of people would say yes.
—Susan Caminiti, “The ordeal of infertility,” Fortune, August 08, 1994
Family balancing has been much in the news of late because of its inherent ethical implications. The biggest issue comes from a technique called preimplantation genetic diagnosis (PGD) in which an egg is harvested, fertilized with sperm, and then screened for certain genetic characteristics. If the embryo has the characteristics the parents want (or if it lacks certain characteristics, such as a genetic disease carried by one or both parents), then it's implanted in the mother's womb and the pregnancy proceeds normally. The problem lies in the opposite case: if the embryo doesn't have the required characteristics (or has the disease). In this case, the embryo is often destroyed and another egg is chosen. This rejection of "unsuitable" embryos makes many people uneasy, particularly when the rejection is based on non-life threatening criteria, such as the embryo's sex.

The good news is that a new technique — called MicroSort (see the first example citation) — has been developed that eliminates embryo destruction. This technique attempts to sort sperm into two groups: those carrying the X-chromosome and those carrying the Y-chromosome. If your family balancing requires a girl, then the X-chromosome sperm are used to fertilize the egg; if a boy is preferred, the Y-chromosome group is sent into action.

Note, too, that an occasional synonym for family balancing is gender balancing (1978) although this phrase is more often used in the context of addressing gender imbalances in workforces or educational materials.